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3/(x-5)+8/x=2

3/(x-5)+8/x=2 equation

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Numerical solution:

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The solution

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  3     8    
----- + - = 2
x - 5   x
3x−5+8x=2\frac{3}{x - 5} + \frac{8}{x} = 2
Simplify
Detail solution
Given the equation:
3x−5+8x=2\frac{3}{x - 5} + \frac{8}{x} = 2
Multiply the equation sides by the denominators:
x and -5 + x
we get:
x(3x−5+8x)=2xx \left(\frac{3}{x - 5} + \frac{8}{x}\right) = 2 x
11x−40x−5=2x\frac{11 x - 40}{x - 5} = 2 x
11x−40x−5(x−5)=2x(x−5)\frac{11 x - 40}{x - 5} \left(x - 5\right) = 2 x \left(x - 5\right)
11x−40=2x2−10x11 x - 40 = 2 x^{2} - 10 x
Move right part of the equation to
left part with negative sign.

The equation is transformed from
11x−40=2x2−10x11 x - 40 = 2 x^{2} - 10 x
to
−2x2+21x−40=0- 2 x^{2} + 21 x - 40 = 0
This equation is of the form
a*x^2 + b*x + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
where D = b^2 - 4*a*c - it is the discriminant.
Because
a=−2a = -2
b=21b = 21
c=−40c = -40
, then
D = b^2 - 4 * a * c = 

(21)^2 - 4 * (-2) * (-40) = 121

Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
x1=52x_{1} = \frac{5}{2}
x2=8x_{2} = 8
The graph
05-10-510152025-2000020000
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
8 + 5/2
52+8\frac{5}{2} + 8 
=
21/2
212\frac{21}{2} 
product
8*5
---
 2
5⋅82\frac{5 \cdot 8}{2} 
=
20
2020 
20
Rapid solution [src] 
x1 = 5/2
x1=52x_{1} = \frac{5}{2} 
x2 = 8
x2=8x_{2} = 8 
x2 = 8
Numerical answer [src] 
x1 = 8.0
x2 = 2.5
x2 = 2.5
The graph
3/(x-5)+8/x=2 equation
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